Papers

We employ a simplified form of Tod's ansatz—which is built on the LeBrun–Tod ansatz—to construct one-sided type-D Ricci-flat multi-centre metrics. These metrics are Hermitian non-Kähler and conformally Kähler, and they are determined by a pair of generating potentials: an axisymmetric harmonic function (different from the one originally proposed by Tod) in an auxiliary 3D flat space and its harmonic conjugate. We carry out a systematic study of these multi-centre metrics, including their rod structure, asymptotic structure and recovering from them various known closed-form examples. In particular, we show that, when fitted into the scheme of multi-soliton solutions on flat space constructed by the present author using the inverse-scattering method, these metrics, with number of centres $n\le 3$, have a free-soliton number 2 or 3 greater than their phantom-soliton number; we conjecture that this is true for general $n$.

We identify a Hermitian curvature flow which preserves HKT geometry, and whose fixed points are HKT-Einstein metrics, equivalent to a flow suggested by Verbitsky in the context of the quaternionic Monge-Ampère equation. We exhibit a fundamental regularity obstruction and a monotonicity formula for the Chern scalar curvature. We formulate a maximal existence time conjecture for this flow, and give a conditional resolution. We establish the existence conjecture in dimension four. We show the existence of a divergent sequence of HKT-Einstein metrics on quaternionic Hopf surfaces. These are the first non-homogeneous examples in the literature, and indicate the delicacy of the convergence question. Finally we classify which strong HKT structures arising from bi-invariant metrics on Lie groups are also HKT-Einstein.

This paper defines Lyra almost Ricci-Bourguignon solitons on twisted warped product manifolds and investigates the interaction between the twisting function, the Lyra scale, and the soliton potential field. We derive the horizontal, vertical, and mixed components of the soliton equation and obtain trace, gradient, and factor-inheritance characterizations. The mixed equation yields new rigidity phenomena: a base-dependent Lyra scale preserves the ordinary separability obstruction, whereas a fiber-dependent scale admits a weighted compensation law that allows genuinely nonseparable twisting functions. We also establish Einstein reductions for conformal, Killing, homothetic, concurrent, and gradient potential fields. Exact flat and non-flat examples, together with local and compact nonexistence results, demonstrate the geometric significance of the scale?twisting interaction.

We compute Simons type equations for parallel mean curvature submanifolds in complex space forms N^n(c), with constant holomorphic sectional curvature c, and product spaces N^n(c)xR. These formulas are then used to characterize some of these submanifolds.

We study Ollivier–Ricci curvature on Cayley graphs of groups of polynomial growth. Our main result shows that non-negative Ollivier–Ricci curvature forces the group to be virtually abelian. As an application, we prove that connected vertex-transitive graphs of polynomial growth and non-negative Ollivier–Ricci curvature are quasi-isometric to $\mathbb{Z}^k$, for some $k\in\mathbb{N}$.

In this paper, we study the fundamental group of compact de cent 4-dimensional spacetimes with lightlike hypersurface curvature. Such manifolds are foliated by totally geodesic null hypersurfaces. We prove that when the leaves of the foliation are compact then the fun damental group of the spacetime is polycyclic. We also prove that a non totally vicious 4-dimensional compact pp-wave spacetime has polycyclic fundamental group.

We study nondegenerate $\mathbb Z_2$ harmonic functions on Euclidean spaces with branching loci given by ellipsoids and planar conics. We first prove that the parameter map associated with Yan's ellipsoidal family is injective. Together with Yan's surjectivity result, this identifies the ellipsoidal models, up to Euclidean motions and overall sign, with nondegenerate harmonic quadratic polynomials of index \(n-1\) and positive critical value. In \(\mathbb R^3\), we use modified ellipsoidal coordinates to construct a nondegenerate $\mathbb Z_2$ harmonic function in a neighborhood of any planar hyperbola. We then show that no global critical $\mathbb Z_2$ harmonic function with a planar hyperbola as branch locus can have finite Almgren frequency at infinity. Using modified paraboloidal coordinates, we construct a global nondegenerate $\mathbb Z_2$ harmonic function with any prescribed planar parabola as branch locus and obtain a precise asymptotic expansion and scale-down limit. Finally, we identify the parabolic model as the infinitesimal two-valued graph potential of a family of special Lagrangian three-folds constructed by Joyce.

For a closed connected Riemannian manifold $(M,g)$, the Gigli–Mantegazza construction pulls back the quadratic Wasserstein metric under the heat kernel embedding $x\mapsto p_t(x,\cdot)\,d\operatorname{vol}_g$. The resulting family $\widetilde{g}_t$ agrees with Ricci flow to first order in $t$, but in general not to second order. We prove that $$ \widetilde{g}_t =g-2t\operatorname{Ric}_g +t^2\left(-\Delta\operatorname{Ric}_g +2\operatorname{Ric}_g^2-\frac{2}{3}\mathcal{Q}_g\right) +O_{C^0}(t^3), $$ where $\mathcal{Q}_g$ is quadratic in the full curvature tensor. The term $-\Delta\operatorname{Ric}_g$ also occurs in the second-order expansion of Ricci flow, so the discrepancy depends pointwise and quadratically on the curvature. In particular, at a Ricci-flat metric that is not flat, Ricci flow is stationary while $\widetilde{g}_t$ is not. The Gromov–Hausdorff distance between the Gigli–Mantegazza and Ricci-flow metrics is $O(t^2)$, and round spheres show that this estimate is sharp.

In this paper, we prove that there is no complete two-sided stable immersed free boundary minimal hypersurface in the closed Euclidean unit ball $\mathbb{B}^{n+1}$ for $n\leq 5$.

We compare two natural Hardy quantizations carried by the unit cosphere bundle of the round two-sphere. Through $S^2\simeq CP^1$, the cosphere bundle is the unit circle bundle of the canonical bundle $O(-2)$, and its Hardy space assembles the section spaces $H^0(CP^1,O(2\ell))$. Through the real-analytic round metric, the imaginary-time exponential map identifies the same cosphere bundle with every Grauert-tube boundary carrying the adapted complex structure, whose Reeb flow is the geodesic flow. Both Hardy spaces are realized in $L^2(SO(3))$, are multiplicity-free under the left action, and select one line in each Peter-Weyl multiplicity space. We compute the normalized overlap of these lines in closed form as $\frac{\sqrt{\binom{2\ell}{\ell}}}{2^\ell}\frac{\sinh^\ell\tau}{\sqrt{P_\ell(\cosh 2\tau)}}$, with $P_\ell$ the Legendre polynomial. The normalized kernel vectors define an explicit equivariant unitary between the two Hardy spaces, and the squared overlaps are the eigenvalues of the positive trace-class operator $\Pi_h\Pi_\tau\Pi_h$. We derive four exact trace series with the Hardy projectors inserted and show that their sum differs from the full flat trace by a distribution whose Abel regularization has cubic growth at every geodesic period. The eigenvalues also give a genus-zero Fredholm determinant of order zero, and truncating the complete large-$\ell$ expansion at any fixed order gives a finite polylogarithmic expression. A Bargmann-Fock calculation identifies the $\ell$-independent prefactor $(2\sinh 2\tau)^{1/4}e^{-\tau/2}$ in the large-$\ell$ asymptotic of the overlap with the Gaussian matrix coefficient of a metaplectic operator comparing the two contact planes.

We study the Fino-Vezzoni conjecture on compact complex manifolds carrying a balanced Bismut torsion-parallel Hermitian metric. We prove that if such a manifold also admits a pluriclosed Hermitian metric, then it admits a Kähler metric.

For $n\leq 18$, we prove that any smooth immersion of the $n$-torus into the closed unit ball in $\mathbb R^q$ has a point at which the spherical average of $\lvert II(v,v)\rvert^2$ is at least $3n/(n+2)$. This answers a question of Petrunin in these dimensions. The proof combines the scalar curvature obstruction for the torus with a conformal Laplacian argument, reducing the problem to a one-dimensional differential inequality. We also show that this reduction cannot yield the result in dimensions $n\geq19$.

In this note, we establish a refined Schwarz lemma for $V$-harmonic maps. Specifically, we prove that if $u$ is a $V$-harmonic map of generalized dilatation of order $\beta $ from a complete Riemannian manifold with Bakry–Émery Ricci curvature bounded below by a constant $-A$ to a Riemannian manifold with sectional curvature bounded above by a negative constant $-B$, then $$ u^*h\le \frac{Ac(\beta )}{BD(\beta )}g \le \frac{A\beta^2}{B}g, $$ where $c(\beta )=\beta^2/(1+\beta^2)$ and $D(\beta )=1-\lfloor 1/c(\beta )\rfloor c(\beta )^2-(1-\lfloor 1/c(\beta )\rfloor c(\beta ))^2$. Equality in the second inequality holds if and only if $\beta =1,\ 1/\sqrt{2},\ 1/\sqrt{3}, \cdots $. Our result improves the previous bounds obtained by Shen (J. Reine Angew. Math., 1984) for harmonic maps and by Chen–Li–Qiu (Nonlinear Anal., 2022) for $f$-harmonic maps. We also present some applications of our main theorem.

In 1986, Gromov asked whether every complete noncompact $n$-dimensional Riemannian manifold with nonnegative Ricci curvature and scalar curvature at least one satisfies: \[ \Vol_g (B(p, R))\le C_{n}R^{n-2} \] for all $p\in M$ and $R>0$. We answer this question affirmatively using the heat-kernel Fisher metric and Nash entropy.

Banach asked in 1932 whether a real Banach space $X$ whose $n$-dimensional subspaces, for some fixed $1<n<\dim X$, are all linearly isometric must be a Hilbert space. Gromov proved the conjecture for even $n$, and subsequent work settled several odd-dimensional cases. We prove the conjecture for every odd $n$, including all previously unresolved cases. Together with Gromov's result for even $n$, this completes Banach's isometric conjecture in the real case. The proof combines principal bundle theory with a Brouwer degree argument.

For a negatively curved surface, magnetic (or twisted geodesic) flows with nonpositive ”magnetic curvature” have a large-scale geometry on the universal cover that retains some features associated with geodesic flows: magnetic boundary and visibility, possibly magnetically flat strips. This has dynamical consequences: orbit-equivalence to the geodesic flow, density of periodic points, transitivity, entropy-expansivity, existence of equilibrium states.

We prove the existence of a genuinely time-dependent Brakke flow starting from $\Gamma_0 \subset \mathbb{R}^{n+1}$ whose associated multiplicity-one varifold is stationary, provided that, at some singular point, the scale-invariant $L^2$ distance of $\Gamma_0$ from an $n$-dimensional plane has sufficiently small limsup as the scale tends to zero. This yields the dynamical instability of $\Gamma_0$, a notion recently introduced by Stuvard and Tonegawa, and hence the non-uniqueness of Brakke flows starting from $\Gamma_0$. A notable feature of our result is that it holds without assuming the uniqueness of tangent cones at the singular point.

We construct a smooth canonically polarized threefold, not biholomorphic to a product of positive-dimensional varieties, whose normalized Kähler-Einstein metric admits a non-integrable infinitesimal Einstein deformation. The same tangent direction is non-integrable as an infinitesimal complex deformation. In fact, the space of infinitesimal Einstein deformations in our example has real dimension 8, its integrable directions form a real 6-dimensional subspace, and every direction outside that subspace is obstructed. This answers both parts of a suitably generalized version of a question posed by Dai, Wang, and Wei in real dimension 6.

For each integer $k \geq 3$ Hermann Karcher identified a complete singly periodic minimal surface $\Xi_k$ (unique up to similarity) with $2k$ ends asymptotic to the union of $k$ planes intersecting equiangularly along a single line and with genus zero in the quotient by a fundamental translation. Writing $\Xi_{k,m}$ for the quotient of $\Xi_k$ by translation through $m \geq 1$ fundamental periods, we study the Morse index and nullity of the subfamilies $\Xi_{k,2}$ and $\Xi_{3,m}$. In the $m=2$ case we prove for all $k \geq 3$ that $\Xi_{k,2}$ has Morse index $4k-3$ and nullity $3$. In the $k=3$ case we prove that there exists a real number $\alpha^* \in (1/3,1/2)$ such that for all $m \geq 1$ the Morse index of $\Xi_{3,m}$ is $6m- 4 \lfloor m\alpha^* \rfloor - 3$ and its nullity is $3$ unless $m\alpha^*$ is an integer, in which case its nullity is $7$. Both proofs exploit the symmetries of each surface and the Dirichlet-to-Neumann map on a half period to reduce the problem to Fourier-analytic computations on the unit circle (after a conformal change).

A well-known conjecture in complex geometry states that a compact Hermitian manifold with constant Chern holomorphic sectional curvature must be Kähler when the constant is nonzero and Chern flat when the constant is zero. The conjecture is known in complex dimension two and in several special classes in higher dimensions. For Hermitian metrics with Bismut-parallel torsion, the non-balanced case and the balanced threefold case were established by Chen–Zheng, while the balanced fourfold case was settled recently by Wang–Zheng. In this article, we prove the nonzero case for balanced Bismut-torsion-parallel Hermitian manifolds in every complex dimension. As a corollary, we confirm that every BTP Hermitian manifold with Chern holomorphic sectional curvature is a nonzero constant, then $g$ is Kähler.

We prove uniqueness in the Lorentzian Calderón problem in a semiglobal setting, comparing a potentially large perturbation of the Minkowski metric against a small one. Our proof uses a reduced amount of data, solving a formally determined version of the problem. In contrast to traditional approaches, it employs neither control-theoretic arguments nor a reduction to a geometric inverse problem via microlocal methods or high-frequency solutions. Instead, it relies on distorted plane waves and weighted $L^2$-estimates.

We prove a rigidity theorem for compact spacelike capillary hypersurfaces in de~Sitter and Minkowski spaces: volume-preserving stability forces total umbilicity when the support is a spacelike totally umbilical hypersurface of nonnegative intrinsic curvature. Using the light-cone model, we construct conformal Killing fields tangent to the support and derive a unified Minkowski-type formula valid in all Lorentzian space forms. The resulting mean-zero functions satisfy an inhomogeneous Jacobi equation and the linearized capillary Robin boundary condition, and form canonical finite-dimensional test families. A finite-trace identity, supplemented in the de~Sitter cases by a nonpositive Dirichlet Green correction, detects the umbilicity defect \(n|h|^2-H^2\) with a definite sign; hence, every non-totally-umbilical hypersurface admits an admissible test function with positive second variation. The construction extends to supports of negative intrinsic curvature, where a unique timelike parameter direction prevents the finite trace from being sign-definite.

Cone structures on differentiable manifolds are fields of cones in the tangent bundle. A cone structure is isotrivial, modeled on a fixed immersed submanifold of the projective space, if at each point the projectivised cone is projectively equivalent to that submanifold. We consider cone structures arising from ordinary differential equations via a canonical construction. We prove that isotrivial cone structures in this class, modeled on generic curves in the $n$-dimensional projective space or ruled surfaces in the three-dimensional projective space, are flat. We also discuss applications to causal geometries in four dimensions and dispersionless Lax systems.

We study whether the topology underlying a Calabi-Yau manifold can support natural geometric positivity or negativity structures. In even complex dimension $n \geq 4$ (assuming $b_2=1$ when $n \geq 6$), we strengthen a theorem of Oguiso-Peternell by proving that a Calabi-Yau manifold is not homeomorphic to a weak Fano $n$-fold. The same obstruction applies to Kähler manifolds with quasi-positive holomorphic sectional curvature. A transformation-group analogue excludes, in particular, symplectic manifolds admitting Hamiltonian circle actions with isolated fixed points. On the negative side, we show that the fundamental group of a Calabi-Yau manifold is not isomorphic to that of a Kähler hyperbolic manifold. Taken together, these results exhibit a common topological rigidity separating Calabi-Yau manifolds from several fundamental classes governed by geometric positivity or negativity.

In this paper, we investigate the semilinear parabolic equation $\partial_t f-\Delta f=a(x,t)f^p$ on a complete Riemannian manifold with Ricci curvature bounded from below. By means of Nash-Moser iteration, we establish Li-Yau type gradient estimates for positive solutions to this equation, where the coefficient function $a$ can be either strictly sign-definite or sign-changing. As an application, we derive Liouville theorems for ancient and eternal solutions on manifolds with nonnegative Ricci curvature, generalizing a number of classical results. Our proof features the incorporation and tuning of two parameters in the auxiliary quantities to accommodate the variable coefficient $a$ and to extend the admissible range of $p$.

We show that every diffeomorphism of $\mathbb{R}^n$ for $n \ge 2$ can be approximated by an automorphism of $\mathbb{C}^n$ in the Whitney $C^k$-topology for any positive integer $k$ using the notion of density property. More precisely we find sufficient conditions for this holomorphic approximation of diffeomorphisms to hold and prove that the split real forms of most linear algebraic groups satisfy these conditions. In the same manner we also show holomorphic approximations of volume-preserving diffeomorphisms for the split real forms of linear algebraic groups equipped with the left-invariant volume form.

We develop an intrinsic calculus for smooth functions of paths of arbitrarily low regularity on smooth manifolds. The regularity of paths is defined in terms of a $p$-th variation tensor along a sequence of partitions, for an arbitrary integer $p$; this tensor is constructed as a local symmetric tensor measure along the path. We define pathwise integrals of closed one-forms along paths with finite $p$-th variation and derive a change of variable formula for smooth functions of such paths. For $p=2$, our results give a manifold version of H. Föllmer's pathwise Itô calculus. Our construction only requires an affine connection on the manifold and may be viewed as a higher-order analogue of L. Schwartz's second-order differential geometry. The connection provides a splitting of higher-order tangent vectors into symmetric tensor components and leads to a geometric transfer principle: the change-of-variable formula defines an intrinsic, connection-independent functional of the reduced $p$-jet of the test function, whose canonical highest-order component is determined by the $p$-th variation tensor. Although our results are purely geometric, they apply to manifold-valued stochastic processes with highly irregular paths and yield a higher-order Itô-type calculus for such processes. We illustrate this calculus for exponential lifts of fractional Brownian motions to Riemannian manifolds and Lie groups.

A novel advective Fisher-Rao metric is introduced for optimization tasks on paths of probability measures governed by the continuity equation. This metric is shown to lead to optimal descent directions. It is then shown that this metric arises naturally from three different perspectives: As the rescaled zero-noise limit of the Fisher-Rao metric on path measures, as the expected value of the second variation of the Freidlin–Wentzell large deviation rate functional, and as the Hessian of the Benamou–Brenier action functional from dynamic optimal transport. We supplement this geometric construction with computational experiments. Here, we demonstrate empirically that the advective Fisher-Rao metric yields the desired optimal fitting of probability densities, whereas the Gauss–Newton method yields optimal fitting of velocity fields.

For an almost contact pseudo-metric manifold we study a variation of the metric in the spirit of Meli-Ngakeu-Olea. Such variation is defined using a vector field on the manifold and the procedure shifts the signature of the metric by increasing the positive index by two. First we derive conditions on the vector field for the obtained manifold to remain almost contact pseudo-metric. Then we study the cases of almost pseudo co-Kähler manifolds. In particular we obtain geometric characterizations of variations for which the fundamental form remains closed and establish a local decomposition into a product. We further investigate the variation in the setting of statistical structures.

We establish a sharp lower bound for the total Gauss-Kronecker curvature of convex hypersurfaces in Cartan-Hadamard manifolds with pinched negative curvature. The bound holds in all dimensions when the diameter is small relative to the curvature scale, and in dimensions 4 and 5 without any restriction on the diameter, provided that the pinching is sufficiently tight. The proofs are based on the Chern-Gauss-Bonnet theorem and weighted Hsiung-Minkowski inequalities. As an application, we obtain the isoperimetric inequality of the Cartan-Hadamard conjecture in dimension 5 under sufficiently pinched curvature.